Suppose a.sub.1 and a.sub.2 are the complex amplitudes of two modes of an electromagnetic field. In Quantum Mechanics, a.sub.1 and a.sub.2, suitably normalized, become the annihilation operators of the modes, and satisfy .vertline.a.sub.i, a.sub.j .vertline.=.delta..sub.i,j, where i,j=1,2 and .delta..sub.ij is the Kronecker delta. (Henceforth, we will express our ideas in quantum mechanical language. In light of this and of the fact that Quantum Mechanics describes both the classical and quantum regimes of any phenomenon, the present invention is intended to apply in both these regimes.) Suppose we denote the average of any operator O by (O), and we let (A,B)=((A-(A))(B-(B)))=(AB)-(A)(B) for any operators A and B. To describe the probabilistic fluctuations of the two modes a.sub.1 and a.sub.2, one may use the covariance parameters [N.sub.1, M.sub.1, N.sub.2, M.sub.2, N.sub.12, M.sub.12 ], which are defined by EQU N.sub.1 =(a.sub.1, a.sub.1), (1) EQU M.sub.1 =(a.sub.1, a.sub.1), (2) EQU N.sub.2 =(a.sub.2, a.sub.2), (3) EQU M.sub.2 =(a.sub.2, a.sub.2), (4) EQU N.sub.12 =(a.sub.1, a.sub.2), (5) EQU M.sub.12 =(a.sub.1, a.sub.2). (6)
N.sub.1 and N.sub.2 will be called the a.sub.1 and a.sub.2 noise intensities (the full intensity for, say, mode a.sub.1, is the noise intensity N.sub.1 plus (a.sub.1).sup.2); N.sub.12 and M.sub.12 will be called the correlation parameters; M.sub.1, M.sub.2 will be called the a.sub.1 and a.sub.2 self-squeezing parameters; M.sub.12 will be called the mutual squeezing parameter. If M.sub.12 .noteq.0 or N.sub.12 .noteq.0, we will say that the modes are correlated; if M.sub.12 =N.sub.12 =0, we will say that the modes are uncorrelated; and the act of transforming uncorrelated modes into correlated ones will be referred to as correlating the modes. If M.sub.1 .noteq.0 or M.sub.2 .noteq.0 or M.sub.12 .noteq.0, we will say that the modes are squeezed; if M.sub.1 =M.sub.2 =M.sub.12 =0, we will say that the modes are unsqueezed; and the act of transforming unsqueezed modes into squeezed ones will be referred to as squeezing the modes. If Z.sub..alpha., for .alpha.=1, 2, 3, 4, is a vector with components a.sub.1, a.sub.2, a.sub.1, a.sub.2 taken in that order, then we define the covariance matrix V for modes a.sub.1 and a.sub.2 so that the .alpha.,.beta. entry of V, call it V.sub..alpha..beta., is (Z.sub..alpha., Z.sub..beta.). Two-mode states with a diagonal covariance matrix have M.sub.1 =M.sub.2 =M.sub.12 =N.sub.12 =0.
We will say that modes a.sub.1, a.sub.2 are in a diagonal 2-mode state if their covariance matrix is diagonal. If also N.sub.1 =N.sub.2 =0, the diagonal 2-mode state is the called the vacuum.
We will say that modes a.sub.1, a.sub.2 are in a self-squeezed 2-mode state if M.sub.1 .noteq.0 or M.sub.2 .noteq.0. A self-squeezed state for which N.sub.12 =M.sub.12 =0 shall be referred to as a basic self-squeezed state or an uncorrelated, self-squeezed state. The prior literature contains numerous discussions of basic self-squeezed states. See, for example, H. P. Yuen, Physical Review A, Vol. 13 pp. 2226-2243 (1976). The electric field E due to mode a.sub.1 may be expressed as EQU E=E.sub.o Re(a.sub.1 e.sup.-.omega.1t)=E.sub.o [a.sub.1r cos (.omega..sub.1 t)+a.sub.1i sin (.omega..sub.1 t)], (7)
where E.sub.o is a constant, .omega..sub.1 is the angular frequency of mode 1, and ##EQU1## It is possible and convenient to choose the time we call zero (time origin) so that M.sub.1 is non-negative. If we do so, then the uncertainties in a.sub.1r and a.sub.1i are given by ##EQU2## and these uncertainties satisfy the following uncertainty principle: EQU .DELTA.a.sub.1r .DELTA.a.sub.1i .gtoreq.1/4. (12)
Common lasers produce unsqueezed light, i.e., light such that M.sub.1 =0 and therefore .DELTA.a.sub.1r =.DELTA.a.sub.1i.
We will say that the modes a.sub.1 and a.sub.2 are in a mutually squeezed 2-mode state if M.sub.12 .noteq.0. Mutually squeezed states which satisfy M.sub.1 =M.sub.2 =N.sub.12 =0 will be referred to as basic mutually squeezed states. The prior literature contains numerous discussions of basic mutually squeezed states. See, for example, B. Yurke, Physical Review A, Vol. 32, pp. 300-310 (1985). Suppose, for example, that modes a.sub.1 and a.sub.2 have angular frequencies .omega..sub.1 =.OMEGA.-.epsilon. and .omega..sub.2 =.OMEGA.+.epsilon., respectively. (This is not the only possibility: in general, the modes a.sub.1 and a.sub.2 need not have different frequencies. They might, for example, belong to two different polarization directions of a beam of monochromatic light). One can express the electric field E due to the superposition of such a pair of modes as EQU E=E.sub.o Re[(a.sub.1 e.sup.i.epsilon.t +a.sub.2 e.sup.-i.epsilon.t)e.sup.-i.OMEGA.t ]=E.sub.o [X.sub.1 cos (.OMEGA.t)+X.sub.2 sin (.OMEGA.t)], (13)
where E.sub.o is a constant, and the quantities, X.sub.1, X.sub.2, called the quadratures, are given by EQU X.sub.1 =Re[(a.sub.1 +a.sub.2)e.sup.i.epsilon.t ], (14) EQU X.sub.2 =Re[(a.sub.1 -a.sub.2)(-i)e.sup.i.epsilon.t ]. (15)
It is possible and convenient to choose the time origin so that M.sub.12 is non-negative. If we do so, and if we assume that M.sub.1 =M.sub.2 =N.sub.12 =0, then the quadratures X.sub.1 and X.sub.2 have uncertainties given by ##EQU3## and these uncertainties satisfy the following uncertainty principle: EQU .DELTA.X.sub.1 .DELTA.X.sub.2 .gtoreq.1/4. (18)
We shall say that two modes a.sub.1 and a.sub.2 are in an N.sub.12 correlated state if N.sub.12 .noteq.0. An N.sub.12 correlated state which also satisfies M.sub.1 =M.sub.2 =M.sub.12 =0 will be referred to as a basic N.sub.12 correlated state or an unsqueezed, correlated state.
We shall call a mutual squeezer with characteristic parameters (or characterized by the complex parameters) [.mu., .nu.] any device which transforms two modes a.sub.1 and a.sub.2 into two new modes a'.sub.1, a'.sub.2 so that: ##EQU4## where .vertline..mu..vertline..sup.2 -.vertline..nu..vertline..sup.2 =1. A mutual squeezer transforms a diagonal 2-mode state into a basic mutually squeezed 2-mode state. For two modes with different frequencies, the mutual squeezer transformation may be accomplished by using a non-degenerate parametric down-converter.
We shall call an a.sub.1 self-squeezer with characteristic parameters (or characterized by the complex parameters) [.mu..sub.1, .nu..sub.1 ] any device which transforms a single mode a.sub.1 into a new mode a'.sub.1 so that: ##EQU5## where .vertline..mu..sub.1 .vertline..sup.2 -.vertline..nu..sub.1 .vertline..sup.2 =1. A self-squeezer for mode a.sub.2 may be defined analogously. An a.sub.1 or an a.sub.2 self-squeezer transforms a diagonal 2-mode state into a basic self-squeezed 2-mode state. Self-squeezing of a.sub.1 (or a.sub.2) may be accomplished by using a degenerate parametric down-converter on a.sub.1 (or a.sub.2). (Note that any 1-mode linear transformation can be interpreted as a self-squeezer. Indeed, if b=.alpha.a+.beta.a and thus b=.beta.*a+.alpha.*a, where .alpha. and .beta. are complex numbers and [a,a]=[b,b]=1, then it is easy to show that .vertline..alpha..vertline..sup.2 -.vertline..beta..vertline..sup.2 =1.)
We shall call a mode coupler with characteristic parameters (or characterized by the complex parameters) [.tau., .rho.] any device which transforms two modes a.sub.1, a.sub.2 into two new modes a'.sub.1, a'.sub.2 so that: ##EQU6## where .vertline..tau..vertline..sup.2 +.vertline..rho..vertline..sup.2 =1. A mode coupler transforms a diagonal 2-mode state into a basic N.sub.12 correlated 2-mode state. Beam splitters (i.e., half-silvered mirrors) are mode couplers which produce transformation Eq. (21) when they act on two modes a.sub.1, a.sub.2 possessing the same frequency but different propagation directions. It is important to note that a conventional beam splitter is incapable of producing transformation Eq. (21) upon two modes a.sub.1 and a.sub.2 if these modes have different frequencies. However, other types of mode couplers, such as acousto-optic modulators [see, for example, Chapter 9 of Waves and Fields in Optoelectronics, (Prentice Hall, 1984) by H. A. Haus], and waveguide mode couplers (ibid., Chapter 7) are capable of doing so.
We shall call an a.sub.1 phase shifter with characteristic angle (or characterized by the real parameter) .theta..sub.1 any device that transforms a mode a.sub.1 into a new mode a'.sub.1 =e.sup.i.theta..sbsp.1 a.sub.1. An a.sub.2 phase shifter can be defined analogously.
Some common optical devices may be modelled by combinations of mode couplers, mutual squeezers, self-squeezers and phase shifters. Next, we shall discuss two particular devices, the laser amplifier and the injection locked laser oscillator, which are specially relevant to the present patent because they arise in prior patents to be discussed below. Our discussion of these two devices follows in part the discussion given by the review article by Y. Yamamoto and H. A. Haus in Reviews of Modern Physics, Vol. 58, pp. 1001-1020 (1986), and references therein.
A laser amplifier takes an initial mode with complex amplitude a to a final mode with complex amplitude c such that ##EQU7## where .phi. is a real number, G.gtoreq.0, and f is a so called noise operator. Assuming that EQU [a,a]=1, (23) EQU [c,c]=1, (24) EQU [a,f]=[a,f]=0, (25)
one gets from Eq. (22) that EQU [f,f]=1-G. (26)
From this last equation, one can conclude the following. For 0.ltoreq.G&lt;1, f/.sqroot.1-G acts as an annihilation operator (like a), so, by Eq. (22), the laser amplifier acts in this case as a mode coupler. For G=1, f acts as a commuting number, so the laser amplifier acts as a phase shifter with characteristic angle .phi.. For G&gt;1, f/.sqroot.G-1 acts as a creation operator (like a), so the laser amplifier acts as a mutual squeezer.
An injection locked laser oscillator takes an initial mode with complex amplitude a to a final mode with complex amplitude c such that ##EQU8## where .phi. and .phi.' are real numbers, G and G' are non-negative real numbers, and f is a noise operator. Assuming Eqs. (23) to (25), one gets from Eq. (27) that EQU [f,f]=(G'+1)-G. (28)
For G&lt;G'+1, f/.sqroot.(G'+1)-G acts as an annihilation operator. For G=G'+1, f acts as a commuting number. For G&gt;G'+1, f/.sqroot.G-(G'+1) acts as a creation operator.
In fact, for G&lt;G'+1, the injection locked laser acts as a self-squeezer followed by a mode coupler. Indeed, according to Eqs. (20) and (21), such a sequence of devices takes an initial mode a to a final mode c such that EQU c=.tau.(.mu.a+.nu.a)+.rho.b, (29)
where EQU .vertline..mu..vertline..sup.2 -.vertline..nu..vertline..sup.2 =.vertline..rho..vertline..sup.2 +.vertline..tau..vertline..sup.2 =1,(30) EQU [a,a]=[b,b]=[c,c]=1, (31) EQU [a,b]=[a,b]=0. (32)
If one defines non-negative real numbers G and G', real numbers .phi. and .phi.', and an operator f by ##EQU9## then it is easy to show that Eq. (28) is satisfied and that G&lt;G'+1.
For G=G'+1, the injection locked laser acts as a self-squeezer. Indeed, Eqs. (29) to (35) apply with .tau.=1 and .rho.=0.
For G&gt;G'+1, the injection locked laser acts as a self-squeezer followed by a mutual squeezer. Indeed, according to Eqs. (19) and (20), such a sequence of devices takes an initial mode a into a final mode c such that EQU c=.mu.(.mu.a+.nu.a)+.nu.b, (36)
where EQU .vertline..mu..vertline..sup.2 -.vertline..nu..vertline..sup.2 =.vertline..mu..vertline..sup.2 -.vertline..nu..vertline..sup.2 =1,(37)
and Eqs. (31) and (32) are satisfied. If one defines non-negative real numbers G and G', real numbers .phi. and .phi.', and an operator f by ##EQU10## then it is easy to show that Eq. (28) is satisfied and that G&gt;G'+1.
In U.S. Pat. No. 4,984,298, Slusher and Yurke describe how to produce a travelling electromagnetic wave comprising a basic mutually squeezed 2-mode state. In their preferred embodiment of the invention, a non-linear medium is placed inside a cavity that has one fully silvered and one partially silvered mirror as ends. The medium is pumped by a coherent light source of angular frequency either .OMEGA. or 2.OMEGA.. Two modes of frequencies .OMEGA..+-..epsilon. (where .epsilon. is a frequency determined by the cavity) occupy a vacuum state upon entering the cavity through the half-silvered mirror end, but they occupy a basic mutually squeezed state upon leaving the cavity through the same end. The preferred embodiment of the invention of Slusher and Yurke or any trivial modification thereof, are incapable of producing 2-mode states which possess a covariance matrix more general (i.e., with more non-zero covariance parameters) than that of a basic mutually squeezed state.
In U.S. Pat. No. 5,113,524, Hirota et al propose two types of devices. In their first type of device, an initial mode is sent through a mode coupler. One of the two modes emerging from the coupler then traverses a self-squeezer. The other mode emerging from the coupler is used to control the phase of the pump to the self-squeezer. This pump phase switches from 0 to .pi. (or vice versa) whenever the phase of the self-squeezer's input mode switches from 0 to .pi./2 (or vice versa). In the language of the present patent, Hirota et al's first type of device consists of a mode coupler followed by a self-squeezer. At its simplest, the invention proposed in the present patent requires the use of at least three devices: one mutual squeezer (which Hirota et al's first type of device doesn't have, except for producing the pump), one self-squeezer and one mode coupler. I claim that using these three devices, one can produce (starting from a diagonal 2-mode state) certain 2-mode states which cannot be produced by using only any two out the three devices.
In the second type of device proposed by Hirota et al, an initial mode is sent through a self-squeezer. The output of the self-squeezer is then injected into a slave laser, which acts as an injection locked laser oscillator. The final (before detection) mode of the device is the output of the slave laser. According to the above discussion of injection locked lasers, Hirota et al's second type of device can be modelled in different parameter regimes by either (1) a self-squeezer followed by a mode coupler, or (2) a self-squeezer, or (3) a self-squeezer followed by a mutual squeezer. (We are using the fact that two successive self-squeezing transformations with possibly a phase shift in between can be combined into a single self-squeezing transformation.) Thus, Hirota et al's second type of device, since it contains only 2 out the 3 types of components required by the present invention, cannot produce certain 2-mode states which can be produced with the present invention. I should also mention that Hirota et al's calculation of the signal to noise ratio (SNR) for their second type of device is seriously flawed. Indeed, Hirota et al claim that their device increases the SNR. Actually, as we prove in Appendix A, their device will always decrease the SNR.
Self-squeezer into mutual squeezer and self-squeezer into mode coupler devices have been considered in publications prior to the patent of Hirota et al. See, for example, the aforementioned review by Yamamoto and Haus where a self-squeezer into mutual squeezer device is considered on page 1009. Mutual squeezer into mode coupler devices have also been considered in the prior literature. See, for example, the paper by B. Yurke et al, Physical Review A, Vol. 33, pp. 4033-4054 (1986). FIG. 5 of the latter paper presents a device in which two modes pass through a mutual squeezer in the form of a degenerate four-wave-mixer. The two output modes of the mutual squeezer then pass through a mode coupler in the form of a Mach-Zehnder interferometer. Even though devices containing 2 out of the 3 components required by the present invention have been discussed in the prior art, the use of devices containing all three components has not been discussed previously, to my knowledge, except in my own publications.
I believe that in my publications: International Journal of Modern Physics B, Vol. 6, pp. 1657-1709, (May 20, 1992), and ibid., Vol. 6, pp. 3309-3325 (1992), I have presented a new process for producing 2-mode states with any of the physically possible 2-mode covariance matrices. Devices proposed by previous workers can achieve 2-mode states with some but not all of these covariance matrices.
The more general states whose production concerns us in the present patent may be used en lieu of, for the same applications as basic mutually squeezed 2-mode states. Basic mutually squeezed 2-mode states have been used to improve the accuracy of interferometry and of absorption spectroscopy. (Interferometry: C. M. Caves, Physical Review D, Vol. 23, pp. 1693-1708 (1981); R. S. Bondurant and J. H. Shapiro, Physical Review D, Vol. 30, pp. 2548-2556 (1984); R. E. Slusher and B. Yurke, U.S. Pat. No. 4,984,298. Spectroscopy: B. Yurke and E. A. Whittaker, Optics Letters, Vol. 12, pp. 236-238 (1987); A. S. Lane, M. S. Reid and D. F. Walls, Physical Review Letters, Vol. 60, pp. 1940-1942 (1988); E. S. Polsik, J. Carri, H. J. Kimble, Vol. 68, pp. 3020-3023 (1992).)